{"id":6725,"date":"2024-07-30T11:01:01","date_gmt":"2024-07-30T03:01:01","guid":{"rendered":""},"modified":"2024-07-30T11:01:01","modified_gmt":"2024-07-30T03:01:01","slug":"\u514b\u7f57\u5185\u514b\u79ef_kronecker\u5b9a\u7406","status":"publish","type":"post","link":"https:\/\/mushiming.com\/6725.html","title":{"rendered":"\u514b\u7f57\u5185\u514b\u79ef_kronecker\u5b9a\u7406"},"content":{"rendered":"

\n <\/path> \n<\/svg> <\/p>\n

\u514b\u7f57\u5185\u514b\u79ef<\/h2>\n<\/p>\n
\n

\u6587\u7ae0\u76ee\u5f55<\/h4>\n
    \n
  • \u514b\u7f57\u5185\u514b\u79ef<\/li>\n
  • \n
      \n
    • \u5b9a\u4e49<\/li>\n
    • \u4f8b\u5b50<\/li>\n
    • \u7279\u6027<\/li>\n
    • \n
        \n
      • \u53cc\u7ebf\u6027\u7ed3\u5408\u5f8b<\/li>\n
      • \u6df7\u5408\u4e58\u79ef\u6027\u8d28<\/li>\n
      • \u514b\u7f57\u5185\u514b\u548c<\/li>\n
      • \u4e0e\u62bd\u8c61\u5f20\u91cf\u79ef<\/li>\n
      • \u4e0e\u56fe\u7684\u4e58\u79ef<\/li>\n
      • \u8f6c\u7f6e<\/li>\n<\/ul>\n<\/li>\n
      • \u77e9\u9635\u65b9\u7a0b<\/li>\n
      • \u5386\u53f2<\/li>\n
      • \u53c2\u8003\u8d44\u6599<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n
        \n

        \u6570\u5b66\u4e0a\uff0c\u514b\u7f57\u5185\u514b\u79ef\u662f\u4e24\u4e2a\u4efb\u610f\u5927\u5c0f\u7684\u77e9\u9635\u95f4\u7684\u8fd0\u7b97.\u514b\u7f57\u5185\u514b\u79ef\u662f\u5f20\u91cf\u79ef\u7684\u7279\u6b8a\u5f62\u5f0f\uff0c\u4ee5\u5fb7\u56fd\u6570\u5b66\u5bb6\u5229\u5965\u6ce2\u5fb7\u00b7\u514b\u7f57\u5185\u514b\u547d\u540d.<\/p>\n<\/blockquote>\n

        \u5b9a\u4e49<\/h3>\n

        A \u2297 B A \\otimes B <\/span><\/span>A<\/span><\/span>\u2297<\/span><\/span><\/span><\/span>B<\/span><\/span><\/span><\/span><\/span> \u5982\u679c A A <\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u662f\u4e00\u4e2a m \u00d7 n m \\times n <\/span><\/span>m<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u7684\u77e9\u9635, \u800c B B <\/span><\/span>B<\/span><\/span><\/span><\/span><\/span> \u662f\u4e00\u4e2a p \u00d7 q p \\times q <\/span><\/span>p<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>q<\/span><\/span><\/span><\/span><\/span> \u7684\u77e9\u9635, \u514b\u7f57\u5185\u514b\u79ef\u5219\u662f\u4e00\u4e2a m p \u00d7 n q m p \\times n q <\/span><\/span>m<\/span>p<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span>q<\/span><\/span><\/span><\/span><\/span> \u7684\u5206\u5757\u77e9\u9635
        A \u2297 B = [ a 11 B \u22ef a 1 n B \u22ee \u22f1 \u22ee a m 1 B \u22ef a m n B ] A \\otimes B=\\left[\\begin{array}{ccc} a_{11} B & \\cdots & a_{1 n} B \\\\ \\vdots & \\ddots & \\vdots \\\\ a_{m 1} B & \\cdots & a_{m n} B \\end{array}\\right] <\/span><\/span>A<\/span><\/span>\u2297<\/span><\/span><\/span><\/span>B<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>B<\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>m<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>B<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u22f1<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>B<\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>m<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>B<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
        \u66f4\u5177\u4f53\u5730\u53ef\u8868\u793a\u4e3a [ 1 ] { }^{[1]} <\/span><\/span><\/span><\/span>[<\/span>1<\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
        A \u2297 B = [ a 11 b 11 a 11 b 12 \u22ef a 11 b 1 q \u22ef \u22ef a 1 n b 11 a 1 n b 12 \u22ef a 1 n b 1 q a 11 b 21 a 11 b 22 \u22ef a 11 b 2 q \u22ef \u22ef a 1 n b 21 a 1 n b 22 \u22ef a 1 n b 2 q \u22ee \u22ee \u22f1 \u22ee \u22ee \u22ee \u22f1 \u22ee a 11 b p 1 a 11 b p 2 \u22ef a 11 b p q \u22ef \u22ef a 1 n b p 1 a 1 n b p 2 \u22ef a 1 n b p q \u22ee \u22ee \u22ee \u22f1 \u22ee \u22ee \u22ee \u22ee \u22ee \u22ee \u22f1 \u22ee \u22ee \u22ee a m 1 b 11 a m 1 b 12 \u22ef a m 1 b 1 q \u22ef \u22ef a m n b 11 a m n b 12 \u22ef a m n b 1 q a m 1 b 21 a m 1 b 22 \u22ef a m 1 b 2 q \u22ef \u22ef a m n b 21 a m n b 22 \u22ef a m n b 2 q \u22ee \u22ee \u22f1 \u22ee \u22ee \u22ee \u22f1 \u22ee a m 1 b p 1 a m 1 b p 2 \u22ef a m 1 b p q \u22ef \u22ef a m n b p 1 a m n b p 2 \u22ef a m n b p q ] . A \\otimes B=\\left[\\begin{array}{cccccccccc} a_{11} b_{11} & a_{11} b_{12} & \\cdots & a_{11} b_{1 q} & \\cdots & \\cdots & a_{1 n} b_{11} & a_{1 n} b_{12} & \\cdots & a_{1 n} b_{1 q} \\\\ a_{11} b_{21} & a_{11} b_{22} & \\cdots & a_{11} b_{2 q} & \\cdots & \\cdots & a_{1 n} b_{21} & a_{1 n} b_{22} & \\cdots & a_{1 n} b_{2 q} \\\\ \\vdots & \\vdots & \\ddots & \\vdots & & & \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{11} b_{p 1} & a_{11} b_{p 2} & \\cdots & a_{11} b_{p q} & \\cdots & \\cdots & a_{1 n} b_{p 1} & a_{1 n} b_{p 2} & \\cdots & a_{1 n} b_{p q} \\\\ \\vdots & \\vdots & & \\vdots & \\ddots & & \\vdots & \\vdots & & \\vdots \\\\ \\vdots & \\vdots & & \\vdots & & \\ddots & \\vdots & \\vdots & & \\vdots \\\\ a_{m 1} b_{11} & a_{m 1} b_{12} & \\cdots & a_{m 1} b_{1 q} & \\cdots & \\cdots & a_{m n} b_{11} & a_{m n} b_{12} & \\cdots & a_{m n} b_{1 q} \\\\ a_{m 1} b_{21} & a_{m 1} b_{22} & \\cdots & a_{m 1} b_{2 q} & \\cdots & \\cdots & a_{m n} b_{21} & a_{m n} b_{22} & \\cdots & a_{m n} b_{2 q} \\\\ \\vdots & \\vdots & \\ddots & \\vdots & & & \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{m 1} b_{p 1} & a_{m 1} b_{p 2} & \\cdots & a_{m 1} b_{p q} & \\cdots & \\cdots & a_{m n} b_{p 1} & a_{m n} b_{p 2} & \\cdots & a_{m n} b_{p q} \\end{array}\\right] . <\/span><\/span>A<\/span><\/span>\u2297<\/span><\/span><\/span><\/span>B<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>p<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>m<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>m<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>m<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>p<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>p<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>m<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>m<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>m<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>p<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u22f1<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u22f1<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>q<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>q<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>p<\/span>q<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>m<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>q<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>m<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>q<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>m<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>p<\/span>q<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u22f1<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22f1<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>p<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>m<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>m<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>m<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>p<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>p<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>m<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>m<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>m<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>p<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u22f1<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u22f1<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>q<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>q<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>p<\/span>q<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>m<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>q<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>m<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>q<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>m<\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>p<\/span>q<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

        \u4f8b\u5b50<\/h3>\n

        [ 1 2 3 1 ] \u2297 [ 0 3 2 1 ] = [ 1 \u22c5 0 1 \u22c5 3 2 \u22c5 0 2 \u22c5 3 1 \u22c5 2 1 \u22c5 1 2 \u22c5 2 2 \u22c5 1 3 \u22c5 0 3 \u22c5 3 1 \u22c5 0 1 \u22c5 3 3 \u22c5 2 3 \u22c5 1 1 \u22c5 2 1 \u22c5 1 ] = [ 0 3 0 6 2 1 4 2 0 9 0 3 6 3 2 1 ] \\left[\\begin{array}{ll}1 & 2 \\\\ 3 & 1\\end{array}\\right] \\otimes\\left[\\begin{array}{ll}0 & 3 \\\\ 2 & 1\\end{array}\\right]=\\left[\\begin{array}{llll}1 \\cdot 0 & 1 \\cdot 3 & 2 \\cdot 0 & 2 \\cdot 3 \\\\ 1 \\cdot 2 & 1 \\cdot 1 & 2 \\cdot 2 & 2 \\cdot 1 \\\\ 3 \\cdot 0 & 3 \\cdot 3 & 1 \\cdot 0 & 1 \\cdot 3 \\\\ 3 \\cdot 2 & 3 \\cdot 1 & 1 \\cdot 2 & 1 \\cdot 1\\end{array}\\right]=\\left[\\begin{array}{cccc}0 & 3 & 0 & 6 \\\\ 2 & 1 & 4 & 2 \\\\ 0 & 9 & 0 & 3 \\\\ 6 & 3 & 2 & 1\\end{array}\\right] <\/span><\/span>[<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span>\u2297<\/span><\/span><\/span><\/span>[<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>\u22c5<\/span><\/span>0<\/span><\/span><\/span><\/span>1<\/span><\/span>\u22c5<\/span><\/span>2<\/span><\/span><\/span><\/span>3<\/span><\/span>\u22c5<\/span><\/span>0<\/span><\/span><\/span><\/span>3<\/span><\/span>\u22c5<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>\u22c5<\/span><\/span>3<\/span><\/span><\/span><\/span>1<\/span><\/span>\u22c5<\/span><\/span>1<\/span><\/span><\/span><\/span>3<\/span><\/span>\u22c5<\/span><\/span>3<\/span><\/span><\/span><\/span>3<\/span><\/span>\u22c5<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span>\u22c5<\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span><\/span>\u22c5<\/span><\/span>2<\/span><\/span><\/span><\/span>1<\/span><\/span>\u22c5<\/span><\/span>0<\/span><\/span><\/span><\/span>1<\/span><\/span>\u22c5<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span>\u22c5<\/span><\/span>3<\/span><\/span><\/span><\/span>2<\/span><\/span>\u22c5<\/span><\/span>1<\/span><\/span><\/span><\/span>1<\/span><\/span>\u22c5<\/span><\/span>3<\/span><\/span><\/span><\/span>1<\/span><\/span>\u22c5<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>9<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

        [ a 11 a 12 a 21 a 22 a 31 a 32 ] \u2297 [ b 11 b 12 b 13 b 21 b 22 b 23 ] = [ a 11 b 11 a 11 b 12 a 11 b 13 a 12 b 11 a 12 b 12 a 12 b 13 a 11 b 21 a 11 b 22 a 11 b 23 a 12 b 21 a 12 b 22 a 12 b 23 a 21 b 11 a 21 b 12 a 21 b 13 a 22 b 11 a 22 b 12 a 22 b 13 a 21 b 21 a 21 b 22 a 21 b 23 a 22 b 21 a 22 b 22 a 22 b 23 a 31 b 11 a 31 b 12 a 31 b 13 a 32 b 11 a 32 b 12 a 32 b 13 a 31 b 21 a 31 b 22 a 31 b 23 a 32 b 21 a 32 b 22 a 32 b 23 ] \\left[\\begin{array}{ll}a_{11} & a_{12} \\\\ a_{21} & a_{22} \\\\ a_{31} & a_{32}\\end{array}\\right] \\otimes\\left[\\begin{array}{lll}b_{11} & b_{12} & b_{13} \\\\ b_{21} & b_{22} & b_{23}\\end{array}\\right]=\\left[\\begin{array}{llllll}a_{11} b_{11} & a_{11} b_{12} & a_{11} b_{13} & a_{12} b_{11} & a_{12} b_{12} & a_{12} b_{13} \\\\ a_{11} b_{21} & a_{11} b_{22} & a_{11} b_{23} & a_{12} b_{21} & a_{12} b_{22} & a_{12} b_{23} \\\\ a_{21} b_{11} & a_{21} b_{12} & a_{21} b_{13} & a_{22} b_{11} & a_{22} b_{12} & a_{22} b_{13} \\\\ a_{21} b_{21} & a_{21} b_{22} & a_{21} b_{23} & a_{22} b_{21} & a_{22} b_{22} & a_{22} b_{23} \\\\ a_{31} b_{11} & a_{31} b_{12} & a_{31} b_{13} & a_{32} b_{11} & a_{32} b_{12} & a_{32} b_{13} \\\\ a_{31} b_{21} & a_{31} b_{22} & a_{31} b_{23} & a_{32} b_{21} & a_{32} b_{22} & a_{32} b_{23}\\end{array}\\right] <\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>3<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>2<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>3<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2297<\/span><\/span><\/span><\/span>[<\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u23a3<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a2<\/span><\/span><\/span><\/span>\u23a1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>3<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>3<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>3<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>3<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>3<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>3<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>2<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>2<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>3<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>3<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>2<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>2<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>3<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>3<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>2<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>2<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>3<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>3<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a6<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a5<\/span><\/span><\/span><\/span>\u23a4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

        \u7279\u6027<\/h3>\n

        \u53cc\u7ebf\u6027\u7ed3\u5408\u5f8b<\/h4>\n

        \u514b\u7f57\u5185\u514b\u79ef\u662f\u5f20\u91cf\u79ef\u7684\u7279\u6b8a\u5f62\u5f0f, \u56e0\u6b64\u6ee1\u8db3\u53cc\u7ebf\u6027\u4e0e\u7ed3\u5408\u5f8b:
        A \u2297 ( B + C ) = A \u2297 B + A \u2297 C  (if  B  and  C  have the same size)  ( A + B ) \u2297 C = A \u2297 C + B \u2297 C  (if  A  and  B  have the same size)  ( k A ) \u2297 B = A \u2297 ( k B ) = k ( A \u2297 B ) ( A \u2297 B ) \u2297 C = A \u2297 ( B \u2297 C ) \\begin{gathered} A \\otimes(B+C)=A \\otimes B+A \\otimes C \\quad \\text { (if } B \\text { and } C \\text { have the same size) } \\\\ (A+B) \\otimes C=A \\otimes C+B \\otimes C \\quad \\text { (if } A \\text { and } B \\text { have the same size) } \\\\ (k A) \\otimes B=A \\otimes(k B)=k(A \\otimes B) \\\\ (A \\otimes B) \\otimes C=A \\otimes(B \\otimes C) \\end{gathered} <\/span><\/span><\/span>A<\/span><\/span>\u2297<\/span><\/span>(<\/span>B<\/span><\/span>+<\/span><\/span>C<\/span>)<\/span><\/span>=<\/span><\/span>A<\/span><\/span>\u2297<\/span><\/span>B<\/span><\/span>+<\/span><\/span>A<\/span><\/span>\u2297<\/span><\/span>C<\/span><\/span> (if <\/span><\/span>B<\/span> and <\/span><\/span>C<\/span> have the same size) <\/span><\/span><\/span><\/span><\/span>(<\/span>A<\/span><\/span>+<\/span><\/span>B<\/span>)<\/span><\/span>\u2297<\/span><\/span>C<\/span><\/span>=<\/span><\/span>A<\/span><\/span>\u2297<\/span><\/span>C<\/span><\/span>+<\/span><\/span>B<\/span><\/span>\u2297<\/span><\/span>C<\/span><\/span> (if <\/span><\/span>A<\/span> and <\/span><\/span>B<\/span> have the same size) <\/span><\/span><\/span><\/span><\/span>(<\/span>k<\/span>A<\/span>)<\/span><\/span>\u2297<\/span><\/span>B<\/span><\/span>=<\/span><\/span>A<\/span><\/span>\u2297<\/span><\/span>(<\/span>k<\/span>B<\/span>)<\/span><\/span>=<\/span><\/span>k<\/span>(<\/span>A<\/span><\/span>\u2297<\/span><\/span>B<\/span>)<\/span><\/span><\/span><\/span>(<\/span>A<\/span><\/span>\u2297<\/span><\/span>B<\/span>)<\/span><\/span>\u2297<\/span><\/span>C<\/span><\/span>=<\/span><\/span>A<\/span><\/span>\u2297<\/span><\/span>(<\/span>B<\/span><\/span>\u2297<\/span><\/span>C<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
        \u5176\u4e2d, A , B A, B <\/span><\/span>A<\/span>,<\/span><\/span>B<\/span><\/span><\/span><\/span><\/span> \u548c C C <\/span><\/span>C<\/span><\/span><\/span><\/span><\/span> \u662f\u77e9\u9635, \u800c k k <\/span><\/span>k<\/span><\/span><\/span><\/span><\/span> \u662f\u5e38\u91cf.
        \u514b\u7f57\u5185\u514b\u79ef\u4e0d\u7b26\u5408\u4ea4\u6362\u5f8b\uff1a\u901a\u5e38, A \u2297 B A \\otimes B <\/span><\/span>A<\/span><\/span>\u2297<\/span><\/span><\/span><\/span>B<\/span><\/span><\/span><\/span><\/span> \u4e0d\u540c\u4e8e B \u2297 A B \\otimes A <\/span><\/span>B<\/span><\/span>\u2297<\/span><\/span><\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> .
        A \u2297 B A \\otimes B <\/span><\/span>A<\/span><\/span>\u2297<\/span><\/span><\/span><\/span>B<\/span><\/span><\/span><\/span><\/span> \u548c B \u2297 A B \\otimes A <\/span><\/span>B<\/span><\/span>\u2297<\/span><\/span><\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u662f\u7f6e\u6362\u7b49\u4ef7\u7684, \u4e5f\u5c31\u662f\u8bf4, \u5b58\u5728\u7f6e\u6362\u77e9\u9635 P P <\/span><\/span>P<\/span><\/span><\/span><\/span><\/span> \u548c Q Q <\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span>, \u4f7f\u5f97
        A \u2297 B = P ( B \u2297 A ) Q A \\otimes B=P(B \\otimes A) Q <\/span><\/span>A<\/span><\/span>\u2297<\/span><\/span><\/span><\/span>B<\/span><\/span>=<\/span><\/span><\/span><\/span>P<\/span>(<\/span>B<\/span><\/span>\u2297<\/span><\/span><\/span><\/span>A<\/span>)<\/span>Q<\/span><\/span><\/span><\/span><\/span><\/span>
        \u5982\u679c A A <\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u548c B B <\/span><\/span>B<\/span><\/span><\/span><\/span><\/span> \u662f\u65b9\u5757\u77e9\u9635, \u5219 A \u2297 B A \\otimes B <\/span><\/span>A<\/span><\/span>\u2297<\/span><\/span><\/span><\/span>B<\/span><\/span><\/span><\/span><\/span> \u548c B \u2297 A B \\otimes A <\/span><\/span>B<\/span><\/span>\u2297<\/span><\/span><\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u751a\u81f3\u662f\u7f6e\u6362\u76f8\u4f3c\u7684, \u4e5f\u5c31\u662f\u8bf4, \u6211\u4eec\u53ef\u4ee5\u53d6 P = Q P=Q <\/span><\/span>P<\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span> .<\/p>\n

        \u6df7\u5408\u4e58\u79ef\u6027\u8d28<\/h4>\n

        \u5982\u679cA\u3001B\u3001C\u548c D D <\/span><\/span>D<\/span><\/span><\/span><\/span><\/span> \u662f\u56db\u4e2a\u77e9\u9635, \u4e14\u77e9\u9635\u4e58\u79efAC\u548c B D B D <\/span><\/span>B<\/span>D<\/span><\/span><\/span><\/span><\/span> \u5b58\u5728, \u90a3\u4e48:
        ( A \u2297 B ) ( C \u2297 D ) = A C \u2297 B D (\\mathbf{A} \\otimes \\mathbf{B})(\\mathbf{C} \\otimes \\mathbf{D})=\\mathbf{A} \\mathbf{C} \\otimes \\mathbf{B D} <\/span><\/span>(<\/span>A<\/span><\/span><\/span>\u2297<\/span><\/span><\/span><\/span>B<\/span><\/span>)<\/span>(<\/span>C<\/span><\/span><\/span>\u2297<\/span><\/span><\/span><\/span>D<\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>A<\/span><\/span>C<\/span><\/span><\/span>\u2297<\/span><\/span><\/span><\/span>B<\/span>D<\/span><\/span><\/span><\/span><\/span><\/span><\/span>
        \u8fd9\u4e2a\u6027\u8d28\u79f0\u4e3a\u201c\u6df7\u5408\u4e58\u79ef\u6027\u8d28\u201d, \u56e0\u4e3a\u5b83\u6df7\u5408\u4e86\u901a\u5e38\u7684\u77e9\u9635\u4e58\u79ef\u548c\u514b\u7f57\u5185\u514b\u79ef.\u4e8e\u662f\u53ef\u4ee5\u63a8\u51fa, A \u2297 B A \\otimes B <\/span><\/span>A<\/span><\/span>\u2297<\/span><\/span><\/span><\/span>B<\/span><\/span><\/span><\/span><\/span> \u662f\u53ef\u9006\u7684\u5f53\u4e14\u4ec5\u5f53 A \\mathbf{A} <\/span><\/span>A<\/span><\/span><\/span><\/span><\/span><\/span> \u548c B \\mathrm{B} <\/span><\/span>B<\/span><\/span><\/span><\/span><\/span><\/span> \u662f\u53ef\u9006\u7684, \u5176\u9006\u77e9\u9635\u4e3a:
        ( A \u2297 B ) \u2212 1 = A \u2212 1 \u2297 B \u2212 1 (\\mathbf{A} \\otimes \\mathbf{B})^{-1}=\\mathbf{A}^{-1} \\otimes \\mathbf{B}^{-1} <\/span><\/span>(<\/span>A<\/span><\/span><\/span>\u2297<\/span><\/span><\/span><\/span>B<\/span><\/span>)<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>A<\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2297<\/span><\/span><\/span><\/span>B<\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

        \u514b\u7f57\u5185\u514b\u548c<\/h4>\n

        \u5982\u679cA\u662f n \u00d7 n n \\times n <\/span><\/span>n<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u77e9\u9635, B\u662f m \u00d7 m m \\times m <\/span><\/span>m<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>m<\/span><\/span><\/span><\/span><\/span> \u77e9\u9635, I k \\mathbf{I}_{k} <\/span><\/span>I<\/span><\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u8868\u793a k \u00d7 k k \\times k <\/span><\/span>k<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>k<\/span><\/span><\/span><\/span><\/span> \u5355\u4f4d\u77e9\u9635, \u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49\u514b\u7f57\u5185\u514b\u548c \u2295 \\oplus <\/span><\/span>\u2295<\/span><\/span><\/span><\/span><\/span> \u4e3a:
        A \u2295 B = A \u2297 I m + I n \u2297 B . \\mathbf{A} \\oplus \\mathbf{B}=\\mathbf{A} \\otimes \\mathbf{I}_{m}+\\mathbf{I}_{n} \\otimes \\mathbf{B} . <\/span><\/span>A<\/span><\/span><\/span>\u2295<\/span><\/span><\/span><\/span>B<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>A<\/span><\/span><\/span>\u2297<\/span><\/span><\/span><\/span>I<\/span><\/span><\/span>m<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>I<\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2297<\/span><\/span><\/span><\/span>B<\/span><\/span>.<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

        \u4e0e\u62bd\u8c61\u5f20\u91cf\u79ef<\/h4>\n

        \u77e9\u9635\u7684\u514b\u7f57\u5185\u514b\u79ef\u5bf9\u5e94\u4e8e\u7ebf\u6027\u6620\u5c04\u7684\u62bd\u8c61\u5f20\u91cf\u79ef.\u7279\u522b\u5730, \u5982\u679c\u5411\u91cf\u7a7a\u95f4 V \u3001 W \u3001 X V \u3001 W \u3001 X <\/span><\/span>V<\/span>\u3001<\/span>W<\/span>\u3001<\/span>X<\/span><\/span><\/span><\/span><\/span> \u548c Y Y <\/span><\/span>Y<\/span><\/span><\/span><\/span><\/span> \u5206\u522b\u5177\u6709\u57fa { v 1 , \u2026 , v m } \u3001 { w 1 , \u2026 , \\left\\{v_{1}, \\ldots, v_{m}\\right\\} \u3001\\left\\{w_{1}, \\ldots,\\right. <\/span><\/span>{
        \n <\/span>v<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>v<\/span><\/span>m<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span>\u3001<\/span><\/span>{
        \n <\/span>w<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, w n } \u3001 { x 1 , \u2026 , x d } \\left.w_{n}\\right\\} \u3001\\left\\{x_{1}, \\ldots, x_{d}\\right\\} <\/span><\/span><\/span>w<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span>\u3001<\/span><\/span>{
        \n <\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>x<\/span><\/span>d<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span><\/span><\/span> \u548c { y 1 , \u2026 , y e } \\left\\{y_{1}, \\ldots, y_{e}\\right\\} <\/span><\/span>{
        \n <\/span>y<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>y<\/span><\/span>e<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span><\/span><\/span>, \u4e14\u77e9\u9635 A A <\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u548c B B <\/span><\/span>B<\/span><\/span><\/span><\/span><\/span> \u5206\u522b\u5728\u6070\u5f53\u7684\u57fa\u4e2d\u8868\u793a\u7ebf\u6027\u53d8\u6362 S : V \u2192 X S: V \\rightarrow X <\/span><\/span>S<\/span><\/span>:<\/span><\/span><\/span><\/span>V<\/span><\/span>\u2192<\/span><\/span><\/span><\/span>X<\/span><\/span><\/span><\/span><\/span> \u548c T : W \u2192 Y T: W \\rightarrow Y <\/span><\/span>T<\/span><\/span>:<\/span><\/span><\/span><\/span>W<\/span><\/span>\u2192<\/span><\/span><\/span><\/span>Y<\/span><\/span><\/span><\/span><\/span>, \u90a3\u4e48\u77e9\u9635 A \u2297 B A \\otimes B <\/span><\/span>A<\/span><\/span>\u2297<\/span><\/span><\/span><\/span>B<\/span><\/span><\/span><\/span><\/span> \u8868\u793a\u4e24\u4e2a\u6620\u5c04\u7684\u5f20 \u91cf\u79ef S \u2297 T : V \u2297 W \u2192 X \u2297 Y S \\otimes T: V \\otimes W \\rightarrow X \\otimes Y <\/span><\/span>S<\/span><\/span>\u2297<\/span><\/span><\/span><\/span>T<\/span><\/span>:<\/span><\/span><\/span><\/span>V<\/span><\/span>\u2297<\/span><\/span><\/span><\/span>W<\/span><\/span>\u2192<\/span><\/span><\/span><\/span>X<\/span><\/span>\u2297<\/span><\/span><\/span><\/span>Y<\/span><\/span><\/span><\/span><\/span>, \u5173\u4e8e V \u2297 W V \\otimes W <\/span><\/span>V<\/span><\/span>\u2297<\/span><\/span><\/span><\/span>W<\/span><\/span><\/span><\/span><\/span> \u7684\u57fa { v 1 \u2297 w 1 , v 1 \u2297 w 2 , \u2026 , v 2 \u2297 w 1 , \u2026 , v m \u2297 w n } \\left\\{v_{1} \\otimes w_{1}, v_{1} \\otimes w_{2}, \\ldots, v_{2} \\otimes w_{1}, \\ldots, v_{m} \\otimes w_{n}\\right\\} <\/span><\/span>{
        \n <\/span>v<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2297<\/span><\/span>w<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>v<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2297<\/span><\/span>w<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>v<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2297<\/span><\/span>w<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>v<\/span><\/span>m<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2297<\/span><\/span>w<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span><\/span><\/span> \u548c X \u2297 Y X \\otimes Y <\/span><\/span>X<\/span><\/span>\u2297<\/span><\/span><\/span><\/span>Y<\/span><\/span><\/span><\/span><\/span> \u7684\u7c7b\u4f3c\u57fa. [2]<\/p>\n

        \u4e0e\u56fe\u7684\u4e58\u79ef<\/h4>\n

        \u4e24\u4e2a\u56fe\u7684\u90bb\u63a5\u77e9\u9635\u7684\u514b\u7f57\u5185\u514b\u79ef\u662f\u5b83\u4eec\u7684\u5f20\u91cf\u79ef\u56fe\u7684\u90bb\u63a5\u77e9\u9635.\u4e24\u4e2a\u56fe\u7684\u90bb\u63a5\u77e9\u9635\u7684\u514b\u7f57\u5185\u514b\u548c, \u5219\u662f\u5b83\u4eec\u7684\u7b1b\u5361\u513f\u79ef\u56fe\u7684\u90bb\u63a5\u77e9\u9635.<\/p>\n

        \u8f6c\u7f6e<\/h4>\n

        \u514b\u7f57\u5185\u514b\u79ef\u8f6c\u7f6e\u8fd0\u7b97\u7b26\u5408\u5206\u914d\u5f8b:
        ( A \u2297 B ) T = A T \u2297 B T (A \\otimes B)^{T}=A^{T} \\otimes B^{T} <\/span><\/span>(<\/span>A<\/span><\/span>\u2297<\/span><\/span><\/span><\/span>B<\/span>)<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>A<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2297<\/span><\/span><\/span><\/span>B<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

        \u77e9\u9635\u65b9\u7a0b<\/h3>\n

        \u514b\u7f57\u5185\u514b\u79ef\u53ef\u4ee5\u7528\u6765\u4e3a\u4e00\u4e9b\u77e9\u9635\u65b9\u7a0b\u5f97\u51fa\u65b9\u4fbf\u7684\u8868\u793a\u6cd5.\u4f8b\u5982, \u8003\u8651\u65b9\u7a0b A X B = C A X B=C <\/span><\/span>A<\/span>X<\/span>B<\/span><\/span>=<\/span><\/span><\/span><\/span>C<\/span><\/span><\/span><\/span><\/span>, \u5176\u4e2d A \u3001 B A \u3001 B <\/span><\/span>A<\/span>\u3001<\/span>B<\/span><\/span><\/span><\/span><\/span> \u548c C C <\/span><\/span>C<\/span><\/span><\/span><\/span><\/span> \u662f\u7ed9\u5b9a\u7684\u77e9\u9635, X X <\/span><\/span>X<\/span><\/span><\/span><\/span><\/span> \u662f\u672b\u77e5\u7684 \u77e9\u9635.\u6211\u4eec\u53ef\u4ee5\u628a\u8fd9\u4e2a\u65b9\u7a0b\u91cd\u5199\u4e3a ( B \u22a4 \u2297 A ) vec \u2061 ( X ) = vec \u2061 ( A X B ) = vec \u2061 ( C ) \\left(B^{\\top} \\otimes A\\right) \\operatorname{vec}(X)=\\operatorname{vec}(A X B)=\\operatorname{vec}(C) <\/span><\/span>(<\/span><\/span>B<\/span><\/span>\u22a4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2297<\/span><\/span>A<\/span>)<\/span><\/span><\/span><\/span>v<\/span>e<\/span>c<\/span><\/span>(<\/span>X<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>v<\/span>e<\/span>c<\/span><\/span>(<\/span>A<\/span>X<\/span>B<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>v<\/span>e<\/span>c<\/span><\/span>(<\/span>C<\/span>)<\/span><\/span><\/span><\/span><\/span>.<\/p>\n

        \u8fd9\u6837, \u4ece\u514b\u7f57\u5185\u514b\u79ef\u7684\u6027\u8d28\u53ef\u4ee5\u63a8\u51fa, \u65b9\u7a0b A X B = C A X B=C <\/span><\/span>A<\/span>X<\/span>B<\/span><\/span>=<\/span><\/span><\/span><\/span>C<\/span><\/span><\/span><\/span><\/span> \u5177\u6709\u552f\u4e00\u7684\u89e3, \u5f53\u4e14\u4ec5\u5f53 A A <\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u548c B B <\/span><\/span>B<\/span><\/span><\/span><\/span><\/span> \u662f\u975e\u5947\u5f02\u77e9\u9635. (Horn & Johnson 1991, Lemma 4.3.1) .<\/p>\n

        \u5728\u8fd9\u91cc, vec \u2061 ( X ) \\operatorname{vec}(X) <\/span><\/span>v<\/span>e<\/span>c<\/span><\/span>(<\/span>X<\/span>)<\/span><\/span><\/span><\/span><\/span> \u8868\u793a\u77e9\u9635 X X <\/span><\/span>X<\/span><\/span><\/span><\/span><\/span> \u7684\u5411\u91cf\u5316, \u5b83\u662f\u628a X X <\/span><\/span>X<\/span><\/span><\/span><\/span><\/span> \u7684\u6240\u6709\u5217\u5806\u8d77\u6765\u6240\u5f62\u6210\u7684\u5217\u5411\u91cf.<\/p>\n

        \u5982\u679c\u628aX\u7684\u884c\u5806\u8d77\u6765, \u5f62\u6210\u5217\u5411\u91cf x x <\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>, \u5219AXB\u4e5f\u53ef\u4ee5\u5199\u4e3a ( A \u2297 B \u22a4 ) x \\left(A \\otimes B^{\\top}\\right) x \\quad <\/span><\/span>(<\/span><\/span>A<\/span><\/span>\u2297<\/span><\/span>B<\/span><\/span>\u22a4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>x<\/span><\/span><\/span><\/span><\/span><\/span> (Jain 1989, 2.8 2.8 <\/span><\/span>2<\/span>.<\/span>8<\/span><\/span><\/span><\/span><\/span> block Matrices and Kronecker Products).<\/p>\n

        \u5386\u53f2<\/h3>\n

        \u5c3d\u7ba1\u6ca1\u6709\u660e\u663e\u8bc1\u636e\u8bc1\u660e\u5229\u5965\u6ce2\u5fb7\u00b7\u514b\u7f57\u5185\u514b\u662f\u7b2c\u4e00\u4e2a\u5b9a\u4e49\u5e76\u4f7f\u7528\u8fd9\u4e00\u8fd0\u7b97\u7684\u4eba\uff0c\u514b\u7f57\u5185\u514b\u79ef\u8fd8\u662f\u4ee5\u5176\u540d\u5b57\u547d\u540d.\u786e\u5b9e\uff0c\u5728\u5386\u53f2\u4e0a\uff0c\u514b\u7f57\u5185\u514b\u79ef\u66fe\u4ee5Johann Georg Zehfuss\u540d\u5b57\u547d\u540d\u4e3aZehfuss\u77e9\u9635.<\/p>\n

        \u53c2\u8003\u8d44\u6599<\/h3>\n
          \n
        1. Van Loan C F. The ubiquitous Kronecker product[J]. Journal of computational and applied mathematics, 2000, 123(1): 85-100.<\/li>\n
        2. Pages 401\u2013402 of Dummit, David S.; Foote, Richard M., Abstract Algebra 2, New York: John Wiley and Sons, Inc., 1999, ISBN 0-471-36857-1<\/li>\n
        3. D. E. Knuth: \u201cPre-Fascicle 0a: Introduction to Combinatorial Algorithms\u201d, zeroth printing (revision 2), to appear as part of D.E. Knuth: The Art of Computer Programming Vol. 4A<\/li>\n<\/ol>\n
          \n

          \u672c\u4eba\u6c34\u5e73\u6709\u9650,\u82e5\u6709\u4e0d\u59a5\u4e4b\u5904, \u6073\u8bf7\u6279\u8bc4\u6307\u6b63.<\/strong>
          \u4f5c\u8005: \u56fe\u7075\u7684\u732b<\/strong><\/p>\n

          \u4f5c\u8005\u90ae\u7bb1: turingscat@126.com<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"\u514b\u7f57\u5185\u514b\u79ef_kronecker\u5b9a\u7406\u6570\u5b66\u4e0a\uff0c\u514b\u7f57\u5185\u514b\u79ef\u662f\u4e24\u4e2a\u4efb\u610f\u5927\u5c0f\u7684\u77e9\u9635\u95f4\u7684\u8fd0\u7b97.\u514b\u7f57\u5185\u514b\u79ef\u662f\u5f20\u91cf\u79ef\u7684\u7279\u6b8a\u5f62\u5f0f\uff0c\u4ee5\u5fb7\u56fd\u6570\u5b66\u5bb6\u5229\u5965...","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[],"tags":[],"_links":{"self":[{"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/posts\/6725"}],"collection":[{"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/comments?post=6725"}],"version-history":[{"count":0,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/posts\/6725\/revisions"}],"wp:attachment":[{"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/media?parent=6725"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/categories?post=6725"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/tags?post=6725"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}